Quantum cluster algebras and representations of shifted quantum affine algebras

Abstract

We construct a new quantization $K_t(\mathcal{O}^{sh}_{\mathbb{Z}})$ of the Grothendieck ring of the category $\mathcal{O}^{sh}_{\mathbb{Z}}$ of representations of shifted quantum affine algebras (of simply-laced type). We establish that our quantization is compatible with the quantum Grothendieck ring $K_t(\mathcal{O}^{\mathfrak{b},+}_{\mathbb{Z}})$ for the quantum Borel affine algebra, namely that there is a natural embedding $K_t(\mathcal{O}^{\mathfrak{b},+}_{\mathbb{Z}})\hookrightarrow K_t(\mathcal{O}^{sh}_{\mathbb{Z}})$. Our construction is partially based on the cluster algebra structure on the classical Grothendieck ring discovered by Geiss-Hernandez-Leclerc. As first applications, we formulate a quantum analogue of $QQ$-systems (that we make completely explicit in type $A_1$). We also prove that the quantum oscillator algebra is isomorphic to a localization of a subalgebra of our quantum Grothendieck ring and that it is also isomorphic to the Berenstein-Zelevinsky's quantum double Bruhat cell $\mathbb{C}_t[SL_2^{w_0,w_0}]$.